The simulations are initialised with stellar wind properties corresponding
to a blue supergiant star. We let the system evolved to a few
\(10^{4}\, \rm yr\)  up to a time \(t_{\rm wr}-\Delta t\) . This time interval corresponds
to the typical timescale of a luminous blue variable phase [1]}
and it permits to have potential magneto-hydrodynamical boundary effects provoked by
the onset of stellar rotation at \(t=0\)  to be transported out of the
computational domain under the effects of the constant wind inflow.
The blue supergiant stellar properties are as follows. We use
\(\dot{M}^{\rm bsg}=10^{-6}\, \rm M_{\odot }\, \rm yr^{-1}\) , which is of
the order of magnitude of that of Sher 25 [2]},
\(v_{\rm w}^{\rm bsg}=500\, \rm km\, \rm s^{-1}\) , that is consistent with
the values for HD 168625 [3]}
and for Sk\(-\) 69279 [4]}.
The stellar radius is \(R_{\star }^{\rm bsg}=20\, \rm R_{\odot }\) , which is that of
\(Sk-69279\)  [4]},
and \(v_{\rm rot}^{\rm bsg}=60\, \rm km\, \rm s^{-1}\)  [3]}.
The surface magnetic field of the blue supergiant ancestor star is taken to be
\(B_{\star }^{\rm bsg}=1\, \rm G\) , which is of the order of the magnetic fields
observed in other cool stars [7]}, [8]}, [9]}.
